For the subjects of this general heading see the articles ALGEBRA; ALGEBRAIC FORMS; ARITHMETIC; COMBINATORIAL ANALYSIS; DETERMINANTS; EQUATION; FRACTION, CONTINUED; INTERPOLATION; LOGARITHMS; MAGIC SQUARE; PROBABILITY.
To Lagrange, perhaps more than to any other, the theory of differential equations is indebted for its position as a science, rather than a collection of ingenious artifices for the solution of particular problems. To the calculus of finite differences he contributed the beautiful formula of interpolation which bears his name; although substantially the same result seems to have been previously obtained by Euler.
The one is a problem of interpolation, the other a step towards the solution of an equation in finite differences.
Reference may also be made to the special articles mentioned at the commencement of the present article, as well as to the articles on Differences, Calculus Of; Infinitesimal Calculus; Interpolation; Vector Analysis.
Especially Ha,upt's note) and does not involve the interpolation of matter by the later redactor of Colossians and Ephesians (Holtzmann, Hausrath' and Bruckner, Reihenfolge d.
To the Jewish interpolation of the 1st century B.C. (about 60-40), i.e.
The attempts made during the 19th century to disprove the Pauline authorship now possess merely an historic interest, nor have the various hypotheses of more or less extensive interpolation won any serious support.
"In the beginning of my mathematical studies, when I was perusing the works of the celebrated Dr Wallis, and considering the series by the interpolation of which he exhibits the area of the circle and hyperbola (for instance, in this series of curves whose common base 0 or axis is x, and the ordinates respectively (I -xx)l, (i (I &c), I perceived that if the areas of the alternate curves, which are x, x 3x 3, x &c., could be interpolated, we should obtain the areas of the intermediate ones, the first of which (I -xx) 1 is the area of the circle.
" Now for the interpolation of the rest, I considered that the denominators I, 3, 5, &c., were in arithmetical progression; and that therefore only the numerical coefficients of the numerators were to be investigated.
Its use in the Eucharist of the undivided Church has been continued since the great schism, although the Eastern Church protests against the interpolation 1 Jevons, Introd.
When attention was called to the interpolation in the 9th century it became one cause of the schism between East and West.
The interpolation really witnessed to a deep-lying difference between Eastern and Western theology.
Matritensis 10041 (begun in the year A.D 94 8), the words are omitted under the heading council of Constantinople but inserted under the heading council of Toledo, in the former MS., above the line and in a later hand, which shows conclusively how the interpolation crept in.
The conditions are thus similar to those which arise in interpolation (q.v.).
In the case of a trapezette, for instance, the data are the magnitudes of certain ordinates; the problem of interpolation is to determine the values of intermediate ordinates, while that of mensuration is to determine the area of the figure of which these are the ordinates.
One method is to construct a table for interpolation of x in terms of u, and from this table to calculate values of x corresponding to values of u, proceeding by equal intervals; a quadrature-formula can then be applied.
53 would be explained if the narrative (which is not in Luke) may be held to be an interpolation by one not familiar with the localities.
It only remains to give due honour to one of the most beautiful of legends, that of the deliverance of Adam's spirit from the nether world by the Christ, the earliest form of which is a Christian interpolation inA poc. Moses, ï¿½ 42 (cp. Malan, Adam and Eve, iv.
In the Septuagint column he drew attention to those passages for which there was no Hebrew equivalent by prefixing an obelus; but where the Septuagint had nothing corresponding to the Hebrew text he supplied the omissions, chiefly but not entirely from the translation of Theodotion, placing an asterisk at the beginning of the interpolation; the close of the passage to which the obelus or the asterisk was prefixed was denoted by the metobelus.
23, an interpolation) for the Passover of the Crucifixion; and the ministry would thus be reduced to a single year.
18 or r9) =01.202.4 [this last is a later interpolation from Eusebius] in the Acts of Pilate.
The next great advance on the Trigonometria artificialis took place more than a century and a half afterwards, when Michael Taylor published in 1792 his seven-decimal table of log sines and tangents to every second of the quadrant; it was calculated by interpolation from the Trigonometria to 10 places and then contracted to 7.
" With desire have I desired to eat this Passover, before I suffer and places the bread after the wine, unless indeed the Pauline interpolation comprises the whole of verse 19.
Interpolation then always has a motive.
Fraudulent interpolation, whether the fraud be pious or otherwise, does occur, but is comparatively rare.
Interpolation is then a voluntary alteration, but in practice it is often hard to distinguish from other changes in which its motive is absent.
Interpolation is sometimes due to an inopportune use of knowledge, as when a quotation or a narrative is made to agree with what the interpolator has read elsewhere.
Spectroscopic Measurements and Standards of Wave-Length.- All spectroscopic measurement should be reduced to wavelengths or wave-frequencies, by a process of interpolation between lines the wave-lengths of which are known with sufficient accuracy.
By means of these, spectroscopists are enabled to measure by interpolation the wave-length of any line they may wish to determine.
Interpolation is easy in the case of all observations taken with a grating.
The most convenient and accurate formula of interpolation seems to be that discovered by J.
If within the range5100-3700A, the constants are determined once for all, the formula seems capable of giving by interpolation results accurate to o 2 A, but as a rule the range to which the formula is applied will be much less with a corresponding gain in the accuracy of the results.
7-9, which is plainly an interpolation, perhaps from the margin, upon the qualifications of episcopoi.
The famous correspondence produced next year in evidence against her at the conference of York may have been, as her partisans affirm, so craftily garbled and falsified by interpolation, suppression, perversion, or absolute forgery as to be all but historically worthless.
Windthorst thereupon raised the question in the Reichstag, but the Prussian government refused to take any notice of the interpolation on the ground that there was no right in the constitution for the imperial authority to take cognizance of acts of the Prussian government.
So, on the other hand, there is no single verse or clause which can be plausibly made out to be an interpolation by Zaid at the instance of Abu Bekr, Omar, or Othman.
I, 2, 9-11; the narrative of E, on the other hand, has in part disappeared owing to the interpolation of later material, in part has been retained in xxiv.
Against the Portuguese claim it is argued that the Villancico corresponding to Joao de Lobeiro's poem is an interpolation in the Spanish text, that Portuguese prose was in a rudimentary stage of development at the period when--ex hypothesi - the romance was composed, and that the book was very popular in Spain almost a century before it is even mentioned in Portugal.
They rejected as an illegitimate interpolation the eternal subject of development, and, instead of one continuing God as the subject of all the predicates by which in the logic the absolute is defined, assumed only a series of ideas, products of philosophic activity.
As the word " interpolation " implies, Hermann did not maintain the hypothesis of a congeries of independent " lays."
The interpolation (if it is one) is probably due to local interests.
But in neither form is it free from later interpolation; and its untrustworthiness is shown by its conflicting with data 1 I is now generally recognized, thanks to Volquardsen and others, that Ephorus is the principal authority followed by Diodorus, except in the chapters relating to Sicilian history.
17-21a suggest that it is a later interpolation, such as writings on church discipline were 1 The same motive occurs in the preface to Irenaeus's treatise, Adv.
We have memoirs relating to the proof of the theorem that every numerical equation has a real or imaginary root, the memoir on the Hypergeometric Series, that on Interpolation, and the memoir Determinatio attractionis - in which a planetary mass is considered as distributed over its orbit according to the time in which each portion of the orbit is described, and the question (having an implied reference to the theory of secular perturbations) is to find the attraction of such a ring.
She is not mentioned in the Iliad or the Odyssey, but in Hesiod (Theogony, 409) she is the daughter of the Titan Perses and Asterie, in a passage which may be a later interpolation by the Orphists (for other genealogies see Steuding in Roscher's Lexikon).
There is not only no room in history for this Median king of the Book of Daniel, but it is also highly likely that the interpolation of "Darius the Mede" was caused by a confusion of history, due both to the destruction of the Assyrian capital Nineveh by the Medes, sixty-eight years before the capture of Babylon by Cyrus, and also to the fame of the later king, Darius Hystaspis, a view which was advanced as early in the history of biblical criticism as the days of the Benedictine monk, Marianus Scotus.
In fine, the interpolation of a Median Darius must be regarded as the most glaring historical inaccuracy of the author of Daniel.
But the unity of thought and atmosphere is such as to show that the work is one whole (subject no doubt to a certain amount of redaction and interpolation), and that the apocalyptic part was composed as an introduction to the rest.
The tale, as told in the 1476 chronicle, is clearly an interpolation, for it comes immediately after a distinct statement that "God had helped the Confederates, and that with great labour they had defeated the knights and Duke Leopold," while the passage immediately following joins on to the former quite naturally if we strike out the episode of the "true man," who is not even called Winkelried.
Assuming this, and rejecting the evidence of the 1476 chronicle as an interpolation and full of mistakes, and that of the song as not proved to have been in existence before 1531, Herr Burkli comes to the startling conclusion that the phalanx formation of the Austrians, as well as the name and act of Winkelried, have been transferred to Sempach from the fight of Bicocca, near Milan (April 27, 1522), where a real leader of the Swiss mercenaries in the pay of France, Arnold Winkelried, reall y met his death in very much the way that his namesake perished according to the story.